State Space Modeling of Nonstationary Time Series and Smoothing of Unequally Spaced Data

  • Genshiro Kitagawa
Part of the Lecture Notes in Statistics book series (LNS, volume 25)


A smoothness prior and state space approach to the modeling of nonstationary or irregular time series is shown and various applications obtained so far are reviewed. The smooth change of parameters that characterizes the stochastic structure of the time series is modeled by stochastic linear difference or differential equation. A simple state space representation of the overall process is derived to facilitate the Kalman filter methodology for state estimation. The integrated likelihood of the model is used to determine the parameters contained in the state space model. Given the best choice of parameters that control the smoothness of the structral change, the smoothing algorithm then yields the estimates of the state vector. Detrending, seasonal adjustment, estimation of gradually changing spectrum, decomposition of time series into several series and outlier problem are shown as examples of the use of this approach. Discrete representation of the sampled continuous process is also shown and smoothing of unequally spaced data is shown for illustration.

Key Words

State space representation Kalman filter fixed interval smoothing smoothness priors likelihood nonstationary time series outlier time varying coefficients 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akaike, H. (1980a). Likelihood and the Bayes procedure, in Bayesian Statistics, J.M. Bernado, M.H. Degroot, D.V. Lindley and A.F.M. Smith, ed., University Press, Valencia, Spain, 141–166.Google Scholar
  2. Akaike, H. (1980b). On the use of the predictive likelihood of a Gaussian model, Ann. Inst. Statist. Math., 32, 311–324.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Brotherton, T. and Gersch, W., (1981). A data analytic approach to the smoothing problem and some of its varaitions, in Proceedings of the 20th IEEE Conference on Decision and Control, 1061-1069.Google Scholar
  4. Gersch, W. and Kitagawa, G. (1982). The prediction of time series with trends and seasonalities, Proc. 21st IEEE Conference on Decision and Control.Google Scholar
  5. Good, I.J. (1965). The estimation of probabilities: An essay on modern Bayesian methods, MIT press, Cambridge.zbMATHGoogle Scholar
  6. Good, I.J. and Gaskins, R.A. (1980). Density estimation and bump-hunting/ by the penalized likelihood method exemplified by scattering and meteorite data, JASA, 75, NO.369, 42–73.MathSciNetzbMATHGoogle Scholar
  7. Jones, R.H. (1980). Maximum likelihood fitting of ARMA models to time series with missing observations, Technometrics, 22, 3, 389–395.MathSciNetzbMATHGoogle Scholar
  8. Jones, R.H. (1981). Fitting a continuous time autoregression to discrete data, in Applied Time Series Analysis II, D.F. Findley, ed., Academic Press, 651-682.Google Scholar
  9. Kitagawa, G. (1980). OUTLAP, An outlier analysis program, Computer Science Monographs, No.15, The Institute of Statistical Mathematics, Tokyo.Google Scholar
  10. Kitagawa, G. (1981). A nonstationary time series model and its fitting by a recursive technique, Journal of Time Series Analysis, 2, 103–116.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Kitagawa, G. (1983). Changing spectrum estimation, to appear in J. of Sound and Vibration, 89, No.4.Google Scholar
  12. Kitagawa, G. and Gersch, W. (1982). A smoothness priors approach to the modeling of nonstationary in the covariance time series, submitted for publication.Google Scholar
  13. Kitagawa, G. and Gersch, W. (1982). A smoothness priors approach to the modeling of time series with trend and seasonality, submitted for publication.Google Scholar
  14. Sage, A.P. and Melsa, J.L. (1971). Estimation theory with applications to communications and control, McGraw-Hill, New York.zbMATHGoogle Scholar
  15. Whittaker, E. (1923). On a new method of graduation. Proceedings Edinborough Math. Society, 41, 63–75.Google Scholar
  16. Whittaker, E. and Robinson, G. (1924). The calculus of observations, A treasure on numerical mathematics, Blackie and Son Limited, London.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Genshiro Kitagawa
    • 1
  1. 1.The Institute of Statistical MathematicsMinato-ku, Tokyo 106Japan

Personalised recommendations